He hurried the phrase “educated at Oxford,” or swallowed it, or choked on it, as though it had bothered him before.

Meaning

He wasn’t comfortable saying he was “educated at Oxford”. He didn’t believe that he was educated there and he doubted that other people would so he tried to make it so other people weren’t sure they heard him right. He had said it before and he rehearsed it like he wanted other people to find out he was lying because it “bothered” him to lie about that.